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1. A problem in Statistics is given to ttyee students whose chances of solving it are

respectively. What is the probability that only one of them solves it correctly ?

LBaEA contains 6 white and 7 black balts; and arrother 9ug g contains 4 white ana s'utact balts- one

ball is drawn to*,m" uug A and without noticing its colour,.is prt in the second Pug

g. A ball is then

drawn from the.second U"ag g. 'Find the probabiity that ttre ball drawn is white in colour'

3. Bag A contains 4 white and 5 black balls, and another 9ug g eontains 6 white and 7 black iufit' O""

ball is drawn fno*'tir" Lug e *J *itftout noticing its colour, is put in $: ft:$ bag B' A ball is then

drawn.fromtl,'.*onab"ag.,Findtheprobabilitytr'atttreballdrawnisblackincolour.

4 Three bags contain T white g red, 9 white 6 red, and 5 white 7 red barls respectively. o* Luu, at

random, is drawn from each bag. Find the probability that all of.them are of the same colour'

11 1

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5. Three bags contain 5 white 8 red, 7 white 6 red and--

drawn fr6m each bag at random' Find the probability6 white 5 lred batls respectively. One ball isthat all the three balls tlrawn are of the same

colour.

6. Two cards are drawn,from,a.well,shuffted:pack of lz-*t one after the other without replacement'

Find the prouauitity that- nn" of these'is,a qu.ttt and the other is a king of opposite colour'

7, A speaks truttl in 6A% afthe,cases and B n g0% of the cases' In what percentagt oi t*t' T"

*"-"

iik;iy to conffadict each other in stating the same fact"

& Two cards are drawn one by one without replacemrcnt from a well shuffled pack of 52 cards' Find the"

frourUitity distribution of the number of aces' - - - I '-'9. i speat<s truth in 65To of the cases and.B in,80'% of the cases. [n what percentage of cases are they

liketry to contradici each other in stating the same fact ? 1-

-

l0.Ganesh,appears for'an interview,161 1ss6r'p6sts A and B; selection for which is independent' The

probability of selection for post o * l and for post B i, |. wnut is the probability ttrat Ganesh is

selected for atleast one of the posts ' u.,,", annrrrer lrqo cn olackballs. one ball

11. One bag contains 6 white and 5 black balls: Another bag contains 5 whitq and 3 I

at random, is transferred from the nrst Ua! to tire ,r"ord'bug and then a ball is drawn from the second

bag. Find the probability that the ball drawn is white'

12. one bag contains 5 white and 6 blacrrs balls. Another bag contains 7 white and 3 black balls' one ball

at random is transferred from the first;il; ril;;cond"bag and then a ball is drawn from the second

bag. Find ttre probability that the ball drawn is white'

ili.If A and B are two independent events such that P(A U B) : 0'5' P(A) : 0'2, find P(B).

l4.BagAcontains6redand5blueballsandanotherbagBcontains5redand8blueballs'Aballisdrawn from the uug e *itt out seeing its colour and it iI put into the bag B. Then a ball is drawn from

bag n at random. FirrO tf,. probability that the ball drawn is blue in colour'

15.In bag A there are 5 white *O S red balls, in Bag B, 7 white and 6 red balls and in bag C' 6 white

and 5 red balls. one ball is taken out url*oorn fiom each bag. Find the probability that all the three

balls are of the same colour'

I

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ii. E r* e and B are given to be independent. Find p@), if it is given thar p(A) = 0.35, p(A u B) = 0.60.

17' A arfr' B throw a die alternately till one of ttrem gets a "6' and wins the games. rioo ,t.i. rrrp."tir.probabilities of wining if A starts the game. ,- urv E(ur{'D' I r

18' The probability of A solving a problem i, f ,ro ttnt of B sotving n * i. what is the probabitity ttrat(i) afleast one of them will solve the problem ?(a) only one of them will solve the problem ? $.,

19. Events A and B are given to be independent. Find p@), if it is given ttrat r1e) : o.i4l,rro , ri ofrl

z)' ?heprobability of Ahitting a,targgtis ! ana tnatof B hining u o l.Theybothtueat the target.Find the probability ttrat(r) atleast one of them will hit the rarget.G7) only one of them will hit the tarlet. .. ._ I2L' A tatr die is tossed t'wice- If the number appearing on the top is less than 3, it is a succerr. noo tn

22' From a lot of 30 bulbs, which includes 6 defective bulbs, a simple of 3 bulbs is drawn at Jo* *itnreplacement. Find the probability distribution or trre rr,rmuer of defective bulbs.

. _ _._.__t23" Two cards are etYl:y:qq9-s,ry9]v €,r.+9gm,-1v!-thour replacement, from a well shuffled pack of 52cards. Find the probability distribuiion of number or aceslz1' Twocards are &rYl successively, withour r9placg1nent, from a well-shuffled pact oi rj .;o;. a;;the probability distribution of'nirmber of spades.

'. , -----' --

25' x is taking up subjTts-Mathernatics'.Physics and chemise q mg examination. His probabilities ofgetting Grade A in these sutjects'are'}'2i03 *o oti.Gtively. Find ttre probability that he gets(r) Grade A in all subjecrs;(tt) Grade A is no subjects;(iu) Grade A in nvo subjects. I ,25' A coin is biased so $at,the !re@ is 3 ,Ileq as- lrtcely to occur as a tail. If the coin is tossed twice, find

z7' Theprobability that a person will get an electric contract is J ano the probability rhat he will not get

plumbing contract it +

.If the protability of gening atreast one contract isthat he will get both ?

2

; , what is the probabiliry

28' Find the probability disribution of number of doublets in three throws or. pJ-Jli;."-'-'' ----'

29' There are two b"ry'.T}t firs! bag contains 4 white and 2 black balls, while the second bag contains

:#TnrltJ-tfTJT A bag Is picked ,p .t '-oo* .ra r u.ir ir or.il-"ir riro the probability

30' A pair of dice is tossed trvice. If the random variable X is defined as the number of doublets, find ft;probability distribution of X.

"' frJ*:ased coin is tossed 6 times. Find using Binomial distribution, the probabitity of_ggtting atleast

537 illiii*'+32. A corrymy'has two plants o manufacnre scooters. Plant-l rnanufacnresT0% of the scooters,and plant-2

manufacunes 3A%. At Plant-tr, 80% of the scooters dre rated of standard quallty and at plant-Z, 90Voof the scooters are rated of standa.ld quality. A scooter is chosen at random and is found to be of standardquality. Find the probability that it h4s come from Plant-2. (c.B.s.E. 2000, 20051

33. An unbiased coin is tossed 8 times. Find, by,using binomial distribution, the probability of getting atleast 3 heads. Q+I.c.B.s:E. zooo)

34. An unbiased coin .is tossed l0 times Find,,by using binomial distribution, the probability of gettingafleast3heads. :;.r (A.I.C.B.S.E:2000)

35. A student is given a test with 8 items of true.false type. If he gets 6 or more items correct, he isdelared a pass. Given that he gu€sses the answel to each item, compute the probability that he will

(C.B.S.E.(Conpn.) 2W36. Using binosdal probability distribution- find the probability of obtaining "less rhan 3 heads" when an

unbiased coin is tossed 6 times. i. ' ,. j (c.8.s.E.200r)37- Usiag binomial probability distribution, find.the probability of obtaining *more than 5 heads" when an

unbiasd coin is tossed 8 times. , / .,' , ., :.: (c.B.s.E 2001)38. An insurance company insqred 3000 scooters,,4(n0 cars and 5000 tnrcks. The probabilities of an accident

involving'a scooter, a car aud,a kuck are 0.021 0.03 and 0.04 respectively. One of the insured vehicles: Ineets with an accident. Find the probability that (D it is a scooter, (fi) it is a car, (rrr) it is a truck.

(C. B. S. E. (Compfr. ) 2001)39. A factory has three machines X,.Y and"Z producing 1000, 2000 and 3000 bolts per day respectively.

The machine X produces 1% defwtre bolts. Y produces 1.5% and Z produces 2% defwlve bolts. Atthe end of a day, a bolt is dmwn at random and is found defective. What is the probability that thisdefectivg boli has been produced by,ttrE machige X ? (c.8.s.8.2004

40. In a bolt,factory;'threemachines,A; B and C manufactire 25; 35 and 40 per cenr of the total boltsmanufacnned. Of their outlr.ut, 5, 4,.and 2 Wt eent are defective respectively. A bolt is drawn atrandom and is found to be defective.

'Find the probability that it was manufactured by either machine: AorC., ,., , ,.i,;... (A.I.C.B.S.E.(Comptt.)2002; C.8.5,8.2006)

41. A company has two plants ,to,manufacture bicycles. The first plant manufactures 60% of fie bicyclesand the srcond plant 40%. 8A,7o'of,the.birycles'are rated of standard quahty of the first plant and 90%of standard quality at,thesecond.plant. ,A bieycle is picked up at random and found to be of standardquatity. Find the probability that it comes from the second plant.

, - : , ! ,,' .".i r,1 ,, ; i -, (C.8.S.8.2003,4.1.C.8.5.8. (Comptt.) 200q€. There are trro identical boxes containing respeetively 4 white and 3 red balls, 3 white and 7 red balls.

A box is chosen at random and,a ball,is drawn from it. If the ball.drawn is white, what is theprobability that it is from thefirst b-qx ? . : . . (C. B. S. E. (Compfr . ) 2003)

43,,From a well sttuffled pack of52,c4rds, 3 cards are drawn one-by-one with replacement. Find tlreprobability distribution of number ofl qu. eens., (4. I. C. B. S. E. (Conptt. ) 2003)

4{.ThreeurnsA,BandCconain6redland4white;2redand6white; audlredand5whiteballsrespectively. An urn is chosen at random and a ball is drawn. If ttre ball drawn is found to be red, findthe probability that the ball was drawn from urn A" (c.B.s.E 2oo4)

-15. Three ums A, B and C contain 4 red and 6 white; 3 red and 5 white; and 2 rd and 4 white ballsrespectively. An urn is chosen at random and a ball is drawn. If ttre ball drawn is found to be red, findthe probability that the ball was drawn from urn A" (c.Bs.E 2A0sl

46. A compary has nvo plants to runufacture T.Vs. The first plant manufacturesT0% of the T.Vs and therest are manufactured by the other plant. 80% of the T.Vs manufaetufed by the first plant aie rated ofstaudard quallty, while that of the second plant only 7O% ue of standard quality. If a T.V. chosen atrandom is found to be of st:ndard quahty, find the probability that it was produced by the fust plant.

(d.r.c.B.s.E 2w47. A pair of dic'b'is tttiown 7 times. If getting the total T is consfuered a success, find the probabilrty of

(l) no zuccess, (ff) at least 6 successes.

48. A can salve 90% of the proble,ms given in a boo( and B can solve orldy 70% problems. What is the

probability that atleast one,of th€n will solve the problem selected at random from the book.

49-Bag A contains 3. white and 2 red balls. Bag B contains 4 white and 5 red balls. One O.ff t ar.*o.i. random from one of the bags,and is found to be red. Find ttre probability that is was drawn from Bag B.

chosen at random,ard,a coin.fo dra$rn a*n it. ffthe selected coin is a gold coin, find the probability

that it was drawn from the second box.

5L.t0% of thb:tools produced by a,machine:are.defective. Find the probability distribution of the number

: of dcfeetive tools in a sample of 3 draws at random. - - -,

54,,A box contains 4 gold and 3 silver soins" Another box conains 3 gold and 5 silver coins. A box is., ' &sco.atrandom,ilrdrs.coin,is dra*n',frEm,it. If,the selected coin is a gold coin, find the probability

that it was drawn from the second box"

, 53. The probability that studqrt,enterihg,a university will graduate is 0.4. Find the probability that out of3 students of the university :

' (i) none will graduate,

(ii) only one will graduate ?

(lll) all will graduate.

54. An insurance Lornpany insured,2000 scooters and 3000 motor cycles. The probability of an accident

involving'a scooter:is 0.01 and thatof a motor cycle is 0'02. An insured vehicle met with an accident.

Find the probability that ttre accidented vehicle was a motor cycle.

5$.,A speals the tiuth S,times orlt of 10 times. A dr€' is 'tossed. He reports that it 'fra$ S.'What fi-ttreprobability that it was actually 5 ?

56,.4' coiu is;tossed 4, times,.Find.the ileon ond vada$ce of the probability distribution of the number oiheads.

57.,A bag conains I white aod,61ed,balls, and a sesond bag contains 4 white and 3 red balls. One of the

. , :bags ,is.picked up 8t, nr rflrldofllland a ball:is ,r,andomly drawn from it, and is found to be white in colour.

r,,, Filtd the probability ttrat the drawn ball was from rhe fust bag.

SE"Fgr'A;.B,.fltrd,G,thelehanessof itreing:iselected as,the flumager of a firm are in the ratio 4'" l:?:: respectively.,,The,respeetiverprobabilities f.orthern.to introduce a radical change in marketing stategy:; .3r€,0.3, 0:& and 0.I5",i'If ,the:change does rtake place, find the probabilrty that it is due to the

appoinnnent of B or C"

S9rTwo cards are drawn successively with replacement from a well-shuffled pack of 52 cards. Find th;mean and variance for the number of aces" r -,

fl). There'are two Bags I and tr. Bag I contains 3 white and 4 black balls and Bag II contains 5 white and

6 black balls. One ball is drawn at random. from one of the bags and is found to be white. Find ttreprobability that is was drawn from Bag I.

61. Two bags A and B contain 2 white, 4 red; and 3 white, 3 red balls respectively. One of the bags is

selected at random and a ball is drawn ftom it. If the selected ball is of white colour, find the

probability that it is drawn ftom bag A.A. Arlurn contains 5 white and 3 red balls. Find ttre probability ditribution of the number of red balls,

with replacements in three draws. "--- -:":" :*,63. In a box containing 50 bulbs 5 are defective. What is the probability ttrat a sample of 5 bulbs will have

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at most 2 defective butb ?

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g. Two persons A and B throw a die alternately till one of them gets a 5 and wins the game. A starts the'

g,arne. Find the respective probabilities of their winning the game.

i' 1

u,. The probability of hining a targer by A is , . tf ne fues 5 times, find the probability that he will hit

at least two times. ', ----,,65. Two cards are drawn successively with replacement from a well shuffled pack of 52 cards. Find the

"* probability distribution of number of jacks'

' *,'Apair of dice is thrown 6 times. Getting a total of 7 on the trvo dice is considered a success. Find the

,, probabiliry of getting :

. (i) atleast'5 successes

i (ir) exactly 5 successes

6.In a class having 60% boys, 5% of the boys and lTVo of the girls have anl'9..of more than 150' A*'student

is selectJd at random and found to have an I.Q. of more than 150. Find the probability that the

selected student is a boY.,,-*,ba class having 70% boys,20% of the boys and LO% of the girls are players. A student is selected

,.--';r*ao* from"trre class and found to be atplayer. Find the probabiliry trnt the selected student is a

, 76. There are rwo bags I and II. Bag I contains 2 White and 4 Red balls and bag tr contains 6 white and

3 Red balls. One ball is drawn at random from one of the bags and is found to be red. Find the

72, Anurn contains 4 red balls and 7 btue balls. Two balls are drawn at random with replacement' Find

the probabilifY of getting

(,) i red balls (ii) 2 blue balls (fiD one red and one blue ball,

73. A card is drawn out from a well shuffled pack of 52 cards. If E is the event "the card drawn out is a

king as a queen, and F is the event "the card"drawn out is a queen or an act". Find the probabitity

P@/D.

74. A andB throw a pair of die by turn. The first to throw 9 is awarded a prize. If A stars the game, show

.9that the probability of A getting the prize is 17 '

75. A man is known to speak truth 3 out of 4 times. He throws a die and reports that it is a 6' Find the

probabilitY that it is actuallY 6.

?6. An insurance company insured 2000 scooter driveis, 3000 car drivers and 4000 truck drivers' The

probabilities of their meeting with an accident reqpectively are 0'04, 0'06 and 0'15. One of the insured

p"rro* meets with an accident. Find the probability that he is a car driver'

,,. An insurance company insued 3000 scooter drivers, 5000 car drivers and 7000 firck drivers "r]r: nrotabilities

of their meeting with an accident respectively are 0'04, 0 .05 and 0' 15 . , One of the insured persons meets

, probability ttnt it was drawn from bag II'

n. Find the mean p and variance d for ttre following probability distribution

with an accident. Find the probability that he is a car driver"

?8. T1a,o cards are drawn simultaneously from a well shuffled pack of 52 cards. Find the mean and standard

deviation of the number of kings.79. ha factory which manufactures bolts, machines A, B and C manufacture respectiv ely 25%,3i5% a{rd

40% of the bolts. Of their output 5, 4 and 2 percent are respectively defective bolts. A bolt is drawn at

random from the total production and is found to be defective. Find the probability that it is manufactured

bythemachineB. , --r-----v,$" I" a factory which manufactures bolts, machines A; B and C, manufacture respectively 35%,25Vo and

4A% of the bolts. Of ttrcir output 5, 4 and.4 percent are respectively defective bolts. A bolt is drawn at

r.andom from the total productionand is found to be defective. Find the probability that it is manufactured

81. In a factory which manufactures bolts,''machines A, B and C, manufacture respectively 35Vo,35% arfr30% of thebolts. Of their output 5, 4 and 3 percent are respectively defective bolts. A bolt is drawn at

random from the total prodgction and is found to be defective. Find the probability that it is manufactured

&. Tree bags contain balls as shown in the table below : '

A bag is chosen at random and trpo balls are drawn from it. They happen to be white and red. What is

the probabilrty ttnt they came from the Itr bag ?

83. Coloured balls are distributed in three bags as shown in the following table :

A bag is se,ected at random and then two:balls are randomly drawn from the selected bag. They happen

to. be black and red: What is the probability that they came from bag I ?

84. Coloured balls are distributed in three bags as shown in the following table :

A bag is selected at random and then trro balls are randornly drawn from the selected bag. They happen

to be white and red. What is the probability ttrat they came from bag II ?

85. A man is known to speak the tnrth 3 outgreater ttun 4. Find the probability ttrat it

of 5 times. He throws a-9], ulO reporrs that it is a numberis actually a number greater than 4. ---* *;

E6. The probabiliry that A hits a target is { ano the probability thar B hirs irshoots at the target, what is ttre proUaUility that(t) the target is hit ?(rr) exactly one of them hits the target ?

' 87' Two cards are drawn simultaneousll lor.sllccessively w-ithout replacement) from a *.u,tum&;;.k;;cards. Find the mean and variance ortrre numuei

"f;;;&. -v^"vu!" r^vur

88' A pair of dice foS--:Y-14 times. ffgqtting a doublet is considered . ru.irr,'firrd the mean and variance

89' From a *ot of 30 bulbs which inrcludes:6^!"r*,i*o , ,r-pr. of 4 bulbs is drawn at random withreplace*ent- Find the mean and variance of trre numbei otoerective burbs.90' In a school' etlt *:J9T- students,.out sf whjch 430 aregirls. It is known that out of 430, l0% of thegirls stdy in class XII' what is the probabitit tlr"a;;:d.nt chosen randomly studies in class )ilI,given that the chosen student is a girl-? -

.,,.,91' Two cards aredrawn simultaneously (or successively witrroutlr.placemen$ from a well-shuffled pack oi52 cards' Find the mean, variance ino standard airii.ti* "r

fte number of aces...

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92' A card from a pack of 52 cards is lost. From the remaining cards of tire pack, trvo cards a,i ora*n itrandom and are found to both clubs. Fi"Jth; ffi;ild;?the lost card being of crubs.

93' From a lot of 10 bulbs, which includes 3 defectives, a sample of 2 bulbs is drawn at random. Find theprobability distribution of the number "r

J"r""ti"";ui#^*94' There are two Bags,.Pag I and Bag tr. Bag I contains + *t it" *a s red balls whiteanother Bag II contains3 white and 7 red balls' bne ball iidra-wn at randorn from one of the bags and it is found to be white. Findthe-probability that it was drawn from Bag I.95' A family has 2 children. Find the probability that both are boys, if it is known that(l) at least one of the children is a boy,

(it) the elder child is a boy.96' Abag contains 4 balls' Tlvo balls are drawn at random, and are found to be white. what is the probability97' on a multiple choice examination with fuee'possible answers (out of which only one is corrert) for each

$H*L:Tstions, what is the prqbabiiii r[;;ffiffi;;ou]d get ro* o;;;correct answers just

98. A random variable X has the following p.D. :x0123P(x)OK?X?KDetermine :

0)x 0DP(x < 3)(iii) P(X > 6) (,v)p(0 < x < 3)

99. Find the probability of throwing atmost 2 sixes in 6 throws of a single die.100' Given three indentical boxes I, ft and m*u..r, "onoirii *o coins. In box I, both coins are grrJ."i"r,in box II' bottr are silver coins ana in box IIr, th";il'r on,ioii uoa one silver coin. A person chooses a boxat random and takes out a coin. what is th; p;;b;liliryiiul ,r," other.oin in tii"io* is also of gold ?

.2rs

5 . Ifeach one ofA and B

453K y2

672lQ '7K2 +K

101. Probilities of solving a specific problem independently by A arid n are ] anO

to solve problem independently, find the probability that(i) the problem is solved

i;D exactty one of them solves the problem. l, - ' )102. 5upp. se 5% of men and 0.25% of women have grey hair. A grey haired person is selected at randonl

Wtrit is the probability of this person being male ? Assume that there are equal number of males ana

females.103. Suppose a girl throws a die: If she gets 5 or 6, she tosses a coin 3 times *A *Lt tfr"

""riLi"in"rO'J.If Jtie gets !,2,3 or 4 she tosses a coin once and notes whether a head or tail is obtained. If she obtained

exactly one head, what is the probability that she tlrew 1,2,3 ot 4 with the die ?

104. How many times must a man toss a fair coin so that the probability of having at least onehead is more

than 80%. , l

105. gllhsstudentsinacollege,itisknownthat6}Voresideinhosteland40Voaredayscholars(notresidingiqr hostel). ftevious year results repor that3}% of all students who reside in hostel attain A grade and

20% of day,echolars attain A grade in their annual examination. At the end of the year,one student is

chosen at random from the colege and he has an A grade. What is the probability that the student is a

hostlier ?

106. fug 1 66ntains 3 red and 4 blnsft balls and bag tr contains 4 red and 5 black balls. Two balls are transferred

at random from bag I to bag tr and then a ball is drawn from bag tr. The ball drawn is found to be red in

colour. Find the probability that the tansferred ball were both black.

107. In a certain college, 47o of boys and lVo of girls are taller than 1.75 metres. Further more, 60% of the

students in the coilege are girli. A student isielected at random from the college and found to be taller

than 1.75 m. Find thi proUibitity that the selected student is a girl. : - -

108. Aspeakstruthin60% ofthecases, whileB n90% ofthecases.Inwhatpercentof casesaretheylikely

to contradict each other in stating the, same fact ? In the cases of contradiction do you think, the

statement of B will carry more weight as he speaks truth in more number of cases than A ?

109, P, speaks tnrth in 70% of thecases and Q in 80%,of the cases. In what percent of cases are they likely

to agree in stating the same fact ?

Do you think, when they agree, means both are speaking truth ? - ,

110. A,speats tuttl in 75% of trcc&sesl while B n90Vo of the cases. In whatpercent of cases are they likely

to contadict each other in stating the same fact ?Do you think that statement of B is true ?

3111. The probabilities of fiilo students e a"a B coming to the school in time are I and

Assuming ttrat the events, 'A coming,ill time' and 'B coming in time' are independent, find th probability

of only one of them coming to the school in time.Write at least one advantage of coming to school in time.

1L2. In a group of 50 scouts in a camp, 30 are well trained in first aid techniques while the remaining are

well trained in hospitality but not.in first aid. Two scouts are selected at random from the group.

Find the probability disribution of number of selected scouts who are well trained in first aid"

Find the mean of the distribution also.Write one more value which is expected from a well trained scout.

113. Often it is taken that a truthful person commands more respect in the society. A man is known to speak

the truth 4 out of 5 times. He throws a die and reports that it is actually a six" Find the probabiliry thal

it is actually a six" Do you also agree that the vaiue truthfuiness leads to more respect in the sociqty-.?

1

t respectively. Ifboth try

I respectivety.